3. THE COSMOLOGICAL CONSTANT

In 1915 Albert Einstein introduced his theory of General
Relativity. Like Newton before him, Einstein's desire was to apply his
theory to cosmology. Einstein embraced the prevailing view at that time
that the universe is static. Therefore, he attempted to find solutions
of the form
= 0. It soon became
apparent that even with
Einstein's theory of gravity, as with Newton's, the gravitational
attraction of the matter in the universe causes a static universe to be
unstable. Furthermore, as can be seen from Eq. (4), the subsequent
requirement of
= 0 implies a negative
pressure such that
p = - /
3. For ordinary stellar matter and gas, this relationship is not
physically reasonable.

To remedy such problems, Einstein modified his original field equations
from Eq. (2) to the more general form

(12)

where is the
cosmological constant mentioned in the previous
section. Equation (12) is the most general form of the field equations
that remains consistent with the physical requirements of a relativistic
theory of gravity. The cosmological constant term, for
> 0,
can be viewed as a repulsive form of gravity that is independent of the
curvature of spacetime. The modern approach is to treat
as a
form of energy present even in empty space - vacuum
energy. [5]
This interpretation implies modifying Eq. (12) to

(13)

In the perfect fluid approximation, this leads to an effective fluid
pressure and energy density given by

(14)

(15)

where pm and
m
are the pressure and energy density
of the matter content of the universe. As Eq. (14) shows, the
cosmological constant contributes a negative term to the pressure in the
universe. This affect of the cosmological constant allowed Einstein to
find a static, albeit unstable, solution for the dynamics of the universe.

Once it became known that the universe is expanding, Einstein discarded
the cosmological constant term having no other physical reason to
include it. However, the possible existence of a non-zero cosmological
constant has been a subject of debate ever since. With the cosmological
constant in the picture, the equations for the dynamics of the universe,
Eqs. (4) and (5), generalize to

(16)

(17)

Besides Einstein's static model of the universe, another interesting,
and important, solution to Eqs. (16) and (17), known as the de
Sitter solution, applies to the case of a spatially flat, empty
universe
(m =
0, pm = 0, k = 0). In this case,
supplies the only contribution to the energy density

(18)

Equation (16) shows that under these conditions the universe would be
accelerating
> 0, and Eq. (17)
shows that the Hubble parameter would be given by

(19)

The de Sitter solution for the cosmic scale factor shows that the effect
of the cosmological constant is to cause the accelerating universe to
expand exponentially with time according to

(20)

Obviously, the universe is not completely empty, but the de Sitter
solution remains important because it is possible for a cosmological
constant term to be sufficiently large as to dominate the dynamics of
the universe. The dominant components of the universe are determined by
the relative values of the corresponding density parameters. Dividing
Eq. (17) by H2 produces the analog of Eq. (7)

(21)

where the sign of k has be separated out. The first term on the
right-hand-side is called the matter term and, in analogy with Eq. (8),
also gives the matter density parameter
m. The
second term is the curvature term and is characterized by the curvature
density parameter
k.
Finally, the last term is known as the vacuum-energy density parameter
.
Thus, in a universe
with a cosmological constant, the primary density parameters are

(22)

The present values of these parameters, together with the Hubble
constant, would determine the dynamics of the universe in this model.
[6]